Definition 1. An ordinal $\alpha > 0$ is a limit ordinal if it is not a successor ordinal, i.e. there is no ordinal $\beta$ such that $\alpha = \beta + 1$.
Proposition 1. Let $\alpha > 0$ be an ordinal. The following statements are equivalent:
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$\alpha$ is a limit ordinal.
- \[\alpha = \sup\{\beta:\beta<\alpha\}.\]
- \[\alpha = \bigcup\alpha.\]
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$\alpha$ has no maximum.
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$\alpha$ is inductive.